Optimizing the spin sensitivity of grain boundary junction nanoSQUIDs -- towards detection of small spin systems with single-spin resolution

We present an optimization study of the spin sensitivity of nanoSQUIDs based on resistively shunted grain boundary Josephson junctions. In addition the dc SQUIDs contain a narrow constriction onto which a small magnetic particle can be placed (with its magnetic moment in the plane of the SQUID loop and perpendicular to the grain boundary) for efficient coupling of its stray magnetic field to the SQUID loop. The separation of the location of optimum coupling from the junctions allows for an independent optimization of the coupling factor $\phi_\mu$ and junction properties. We present different methods for calculating $\phi_\mu$ (for a magnetic nanoparticle placed 10\,nm above the constriction) as a function of device geometry and show that those yield consistent results. Furthermore, by numerical simulations we obtain a general expression for the dependence of the SQUID inductance on geometrical parameters of our devices, which allows to estimate their impact on the spectral density of flux noise $S_\Phi$ of the SQUIDs in the thermal white noise regime. Our analysis of the dependence of $S_\Phi$ and $\phi_\mu$ on the geometric parameters of the SQUID layout yields a spin sensitivity $S_\mu^{1/2}=S_\Phi^{1/2}/\phi_\mu$ of a few $\mu_{\rm{B}}/\rm{Hz^{1/2}}$ ($\mu_B$ is the Bohr magneton) for optimized parameters, respecting technological constraints. However, by comparison with experimentally realized devices we find significantly larger values for the measured white flux noise, as compared to our theoretical predictions. Still, a spin sensitivity on the order of $10\,\mu_{\rm B}/\rm{Hz^{1/2}}$ for optimized devices seems to be realistic.